Determine the smallest perfect square whose base ten representation begins with 2011 (reading left to right). How about the smallest positive perfect cube whose base ten representation begins with 2011 (reading left to right)?

*** Calculators are allowed, but no computer programs.

first you are looking for 4 digits of precision so the solution is likely to be close to 4 digits.

the square root of 2011 is about 44.84417465
the square root of 20110 is about 141.8097317

Now just check one digit at a time.  Starting with 3 seems prudent.
141^2=19881 142^2=20164
448^2=200704 449^=201601
1418^2=2010724 1419^2=2013561
4484^2=20106256 4485^2=20115225 bingo!

the cube of 2011 is about 12.62226683
the cube root of 20110 is about 27.19384953
the cube root of 201100 is about 58.58737279

and checking as before (actually this time I left in the decimal and  checked each cube root until I got a solution.
They are:
12.623^3 = 2011.350448
27.194^3 = 20110.33382
58.59^3 = 201127.0548
which means the solution is
5859^3 = 2011270548

Posted by Jer on 2011-09-16 12:45:25